Luke and Leia are trapped inside a trash compactor on the Death Star. The side walls are moving apart at 0.1 m/sec, but the end walls are moving together at 0.3 m/sec. The volume of liquid inside the compactor is 20 cubic meters, a constant. Write an equation expressing the rate of change of depth of the liquid in terms of the width and length of the region inside the compactor.
I only think of
V=wlh=20 .... I'm stuck...can someone please help?
Any help will be appreciated.
yes
V = w L h = 20
dw/dt = .2 m/s
dL/dt = -.6 m/s
h = 20/(wL)
dh/dt = dh/dL dL/dt + dh/dw dw/dt
dh/dL = -20w/(w^2L^2) = -20/(wL^2)
dh/dw = -20/(w^2 L)
so
dh/dt = [-20/(wL^2)](-.6) +[-20/(w^2L)](.2)
simplify that
Well, it seems like Luke and Leia are in quite a tight spot! Let's see if we can help them out with some math humor.
First, let's break down the problem. We know that the volume of the liquid inside the compactor is constant at 20 cubic meters, which means that the rate of change of the volume with respect to time is zero.
We are given that the side walls are moving apart at 0.1 m/sec, so the rate of change of the width (w) with respect to time is 0.1 m/sec. Similarly, the end walls are moving together at 0.3 m/sec, so the rate of change of the length (l) with respect to time is -0.3 m/sec.
To find the rate of change of the depth (h) of the liquid, we can differentiate the volume equation V = wlh = 20 with respect to time (t). Using the product rule, we get:
0 = w * dh/dt * l + h * dw/dt * l + w * h * dl/dt
Since the volume is constant, dh/dt = 0. Rearranging the equation, we have:
h * dw/dt * l = -w * h * dl/dt
Now, to express the rate of change of the depth (dh/dt), we can divide the equation by w * l, cancel out h and solve for dh/dt:
dh/dt = (-w * h * dl/dt) / (w * l)
Simplifying further, we get:
dh/dt = -dl/dt * h / l
And there you have it! The equation expressing the rate of change of the depth of the liquid inside the compactor in terms of the width (w) and length (l) is:
dh/dt = -dl/dt * h / l
Now, let's hope that Luke and Leia can figure out a way to escape that trash compactor before things get too messy!
To find the equation expressing the rate of change of depth of the liquid, we need to consider the relationship between the volume and the dimensions of the region inside the compactor.
Let's assume the width of the region is "w", the length is "l", and the depth of the liquid is "h".
Given: V = wlh = 20
To find the rate of change of depth, we need to differentiate this equation with respect to time.
Differentiating with respect to time (t):
dV/dt = (d/dt)(wlh) = (dw/dt)(lh) + (dl/dt)(wh) + (dh/dt)(wl)
Now we need to find the rates of change of the width (dw/dt) and length (dl/dt) in terms of the velocities of the walls and the rates of change of depth (dh/dt):
Given velocities:
The side walls are moving apart, so dw/dt = 0.1 m/sec (negative because it's moving away from each other).
The end walls are moving together, so dl/dt = -0.3 m/sec.
dh/dt represents the rate of change of depth, which we want to find.
Now, substituting the given values into the equation:
dV/dt = (0.1)(lh) + (-0.3)(wh) + (dh/dt)(wl)
Since the volume V is constant (20 cubic meters), the rate of change of volume (dV/dt) is zero.
0 = (0.1)(lh) + (-0.3)(wh) + (dh/dt)(wl)
Now, we can rearrange the equation to isolate and solve for dh/dt:
0 = (0.1)(lh) + (-0.3)(wh) + (dh/dt)(wl)
Rearranging the equation:
(-0.1)(lh) - (-0.3)(wh) = (dh/dt)(wl)
Now, we can divide both sides by (wl) to solve for the rate of change of depth (dh/dt):
((-0.1)(lh) - (-0.3)(wh))/(wl) = dh/dt
Simplifying the equation further:
(-0.1h + 0.3w) / (w) = dh/dt
Therefore, the equation expressing the rate of change of depth of the liquid in terms of the width (w) and length (l) of the region inside the compactor is:
dh/dt = (-0.1h + 0.3w) / (w)
To solve this problem, we can start by expressing the volume of the liquid in terms of the width (w), length (l), and depth (h) of the region inside the compactor.
Given that the volume of liquid inside the compactor is constant at 20 cubic meters, we can write the equation as:
V = w * l * h = 20
Now, we need to find the rate of change of the depth of the liquid with respect to time. Let's call this rate of change dH/dt, where H is the depth of the liquid and t is time.
To find the relationship between H, w, and l, we can use similar triangles. Consider a small section of liquid at the bottom with width dx. The depth of this section is dH and its height is dh.
Using similar triangles, we can say that:
dh / dH = w / l
Rearranging the equation, we can express dh in terms of dH, w, and l:
dh = (w / l) * dH
To find the rate of change of the depth of the liquid (dH/dt), we can differentiate both sides of the equation with respect to time (t):
d(h) / dt = (w / l) * d(H) / dt
Since w, l, and V are all constants, we can substitute w * l = V into the equation:
d(h) / dt = (V / l) * d(H) / dt
Finally, substituting V = 20, we get the equation expressing the rate of change of the depth of the liquid in terms of the width and length of the region inside the compactor:
d(h) / dt = (20 / l) * d(H) / dt
Therefore, the equation expressing the rate of change of the depth of the liquid in terms of the width and length of the region inside the compactor is:
d(h) / dt = (20 / l) * d(H) / dt